通过正则商求解的相对Dolbeault几何Langlands问题
摘要
设为一个具有abelian regular centralizer且无type N根的affine homogeneous spherical variety。在本文中，我们为提出Dolbeault setting下的relative geometric Langlands conjecture。更具体地，我们 conjectured一个Fourier-Mukai二义性，其Dolbeault period sheaf与一个与Ben-Zvi、Sakellaridis和Venkatesh的dual symplectic representation的polarization closely resemble Dirac-Higgs bundle的sheaf的构建密切相关。这些conjecture可视为Hitchin关于symmetric spaces的brane duality的generalization。我们在多个情形下验证了这些conjecture，包括Friedberg-Jacquet情形、Jacquet-Ichino情形、Rankin-Selberg情形以及Gross-Prasad情形。我们的主要工具是正则商的理论，该理论在HM24中针对symmetric spaces进行了描述。
引用
@article{arxiv.2409.15691,
title = {Relative Dolbeault Geometric Langlands via the Regular Quotient},
author = {Thomas Hameister and Zhilin Luo and Benedict Morrissey},
journal= {arXiv preprint arXiv:2409.15691},
year = {2025}
}
备注
v2: Added assumptions on flatness of regular centralizers, and corrected 2 critical errors in main statements: the placement of the exterior algebra in the Dirac-Higgs bundle in Conjecture 1.9 and the statement of our main global result, Conjecture 1.10, which is a weaker version of Conjecture 1.9 that is proven in examples. Sections 2, 4, and 5 have been heavily re-written